Low-order HDSL2 transmit filter

ABSTRACT

The present invention comprises a method of obtaining coefficients for a transmit filter such that the power spectral density of the output for the different frequencies comes close to but does not exceed a maximum power spectral density of a communication(s) standard. By first doing a convex optimization procedure to obtain the autocorrelation coefficients for the filter and then using the autocorrelation coefficients to determine the filter coefficients, a low-order filter that closely approximates the desired output power spectral densities can be produced.

BACKGROUND OF THE INVENTION

The present invention relates to a transmit filter for high-bit-ratedigital subscriber line (HDSL2 systems). The HDSL2 standard (currently,Draft HDSL2 Standard, T1E1. 4/2000-006, ANSI) requires both the upstreamand the downstream transmitted power spectrum densities (PSDs) to bebelow a certain level. Such transmitted power specifications arestandard on present and future communication systems. It is desired tohave a transmit power that comes close to but does not exceed themaximum allowable power at any frequency. Because the specification isrelatively complex, it is difficult to come up with a low-order filterthat allows the system to approach the maximum allowed power output.

Additionally, transmit filters which are produced using conventionalmethods to approach the maximum PSDs for the different frequencies ofthe HDSL2 specification tend to be quite complex having coefficientsinto the hundreds. This becomes quite difficult to implement in arealistic system. Typically, less accurate lower-order filters are used,giving up some of the allowed power spectrum density at certainfrequencies.

It is desired to have an improved transmit filter and method forcalculating transmit filter coefficients.

SUMMARY OF THE PRESENT INVENTION

This invention describes a two-step procedure for the design of transmitfilter for the communication system with maximum allowable powerspectral density. First, the relevant section of the transmit path ispartitioned into two parts: (i) the transmit filter that needs to bedesigned, and ii) the remaining elements that affect the output but notpart of the transmit filter, modeled as a fixed weighting function. Incontrast to conventional approaches, in one embodiment of the presentinvention, the transmit filter is selected by optimizing theautocorrelation coefficients via convex optimization methods, and thenextracting filter coefficients from autocorrelation coefficients. Thestructure of the convez optimization allows for a Linear Programming(LP) solution. In the system of the present invention, transmit filterwith thirty-two coefficients provides a power spectral density close tothe maximum allowed.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a table illustrating the upstream and downstream maximumpower spectral density for the HDSL2 system, as specified in the HDSL2standard.

FIG. 1B is a diagram of the maximum power spectral density for the HDSL2system.

FIG. 2 is a diagram of the transmit path for the HDSL2 system.

FIG. 3 is a simplified transmit path emphasizing the transmit filter.

FIG. 4 is a flowchart illustrating the method of the present invention.

FIG. 5 are equations used in the convex optimization procedure.

FIG. 6 is a diagram of one embodiment of the magnitude response of aninterpolation filter (IF) which uses one embodiment of the presentinvention.

FIG. 7 is a diagram of the response of a sample-and-hold (SH) elementused in one embodiment of the present invention.

FIG. 8 is a diagram of the frequency response of the analog filters (AF)used in one embodiment of the present invention

FIG. 9 is a diagram of the combined (IF, SH and AF) frequency responseW(f) in one embodiment of the present invention.

FIG. 10 is a table illustrating the transmit filter coefficients for oneembodiment of the present invention.

FIG. 11 is a diagram that illustrates the downstream PSD.

FIG. 12 is a diagram that illustrates the upstream PSD.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

FIGS. 1A and 1B illustrate the PSD limitations of the HDSL2 system asspecified in the HDSL2 standard. The present invention can also be usedwith other DSL specifications or with non-DSL communicationsspecifications. FIG. 1A is a table illustrating the upstream anddownstream maximal power spectral density. FIG. 1B is a graphillustrates both the upstream and downstream maximal PSDs. Note that theupstream PSD has a peak power at around 230 KHz and the downstream PSDhas a peak power at around 350 KHz. The peaks of the upstream anddownstream PSDs correspond to corresponding dips in the downstream andupstream PSDs, respectively, to avoid interference between the upstreamand downstream signals.

FIG. 2 is a diagram of relevant elements of a DSL transmitter in oneembodiment of the present invention. Up-sampler 22 up-samples the inputsignal by a factor of 2 using zero filling. The transmit filter 24filters the signal such that the ultimate output is within the powerspectral density of the specification. An up-sampler 26 then up-samplesthe system by a factor of 2 again, using zero filling. The digitallow-pass filter 28 or interpolation filter (IF) then filters the outputof the up-sampler 26. Sample-and-hold unit 30 is placed after thedigital interpolation filter (IF) and models the effects of theDigital-to-Analog Converter (DAC). Finally, the analog low-pass filter(AF) 32 filters the output of the sample-and-hold (SH) to produce theoutput of the HDSL2 unit. The effects of all filters except the transmitfilter can be modeled as a fixed frequency weighting function, W(f).

FIG. 3 is a simplified diagram with fixed elements 34 before thetransmit filter and fixed elements 36 after the transmit filter. As willbe described below, the system will meet the PSD requirements as long as|W(f)H(f)|²≲S_(max)(f), where S_(max)(f) is the maximum allowable power.

FIG. 4 is a flow chart that illustrates the system of the presentinvention. In step 40, a communication system specification is obtained.This communication system specification will have a power outputlimitations for different frequencies. In step 41, the transmit path ispartitioned into the transmit filter and the other elements. In step 42,in a convex optimization procedure described below, the autocorrelationcoefficients of the transit filter are optimized. In step 44, the filtercoefficients are determined from the autocorrelation coefficients. Aspectral factorization is preferably used to obtain the filtercoefficients from the autocorrelation coefficients. Note that this forthe HDSL2 systems the procedures can be done both for the upstream anddownstream filters. The convex optimization procedure is described asfollows:

Let S_(max)(f) denote the specified maximum allowable power in Watts/Hz.Then

|W(f)H(f)|² ≲S _(max)(f),

for all frequencies. W(f) can be thought of-as a frequency domainweighting function for the contributions of the other elements in thetransmit path besides the transmit filter.

For the example described below${{S_{x_{7}}(f)} = \left. \frac{2}{R_{L}f_{s}} \middle| {{H_{a}(f)}{H_{s\quad h}(f)}{H_{1}(f)}{H(f)}} \middle| {}_{2} \middle| {X_{1}(f)} \middle| {}_{2}{\leq {S_{\max}(f)}} \right.},$

for all frequencies. And${{W(f)}\overset{\Delta}{=}{\sqrt{\frac{2}{R_{L}f_{s}}}{H_{a}(f)}{H_{s\quad h}(f)}{H_{1}(f)}{X_{1}(f)}}},$

is the combined scaled frequency response of the transit path except thetransmit filter. As will be described below, FIG. 9 shows the magnituderesponse of W(f) where we have assumed that |X(f)|²=1; i.e., X₁(f) iswhite which is true at the output of a precoder.

Now, we describe the optimization procedure for selecting the filtercoefficients.

Let, r(nT₂) be the autocorrelation coefficients associated with thefilter impulse response h(nT₂), i.e.,${{r\left( {n\quad T_{2}} \right)} = {\sum\limits_{m = {{- N_{tap}} + 1}}^{N_{tap} - 1}{{h\left( {m\quad T_{2}} \right)}{h\left( {\left( {m + n} \right)T_{2}} \right)}}}},$

for n=−N _(tap)+1,−N _(tap)+2, . . . ,N _(tap)−1. Let${r\overset{\Delta}{=}\left\lbrack {{r(0)}\quad {r\left( T_{2} \right)}\quad \ldots \quad {r\left( {\left( {N_{tap} - 1} \right)T_{2}} \right)}} \right\rbrack^{T}},$

be the autocorrelation vector. Then, we can write${\left| {H(f)} \right|^{2} = {{r(0)} + {\sum\limits_{n = 1}^{N}{2{r(n)}{\cos \left( {2\pi \quad f\quad n\quad T_{2}} \right)}}}}},$

where r(n) is the nth component of the vector r. Note that r(0) is thetotal power under the magnitude response |H(f)|². Thus, the filterdesign problem can be recast as the following optimization problem

max_(r) r(0), such that |W(f)|² |H(f)|² ≲S _(max)(f).

The optimization problem can be written as finite dimensional linearprogramming (LP) if we use a discrete approximation of the upper boundconstraint. We choose a set of N uniformly sampled frequencies${f_{k} = \frac{k\quad f_{s}}{2N}},{k = 0},1,\ldots \quad,{N - 1},$

and replace the upper bound constraint for all frequency by N inequalityconditions as

|W(f _(k))|² |H(f _(k))|² ≲S _(max)(f _(k)), k=0,1, . . . ,N−1

For sufficiently large N this discretization yields a good approximationto the original upper bound condition. Using this discretization werewrite the optimization problem as

max _(r) r(0),

such that |W(f _(k))|² |H(f _(k))|² ≲S _(max)(f _(k)), k=0, 1, . . . ,N−1.

To write the above problem as an LP, we define an N×N_(θ<)matrix F, asize N diagonal matrix W and two vectors s (N×1) shown in the matricesof FIG. 5.

Using these matrices, we write the optimization problem as an LP

min_(r) c ^(T) r, subject to Ar≲b

where ${A\overset{\Delta}{=}\begin{bmatrix}{W\quad F} \\{{- W}\quad F}\end{bmatrix}},{b\overset{\Delta}{=}\begin{bmatrix}s \\0\end{bmatrix}},$

and 0 is a zero vector of compatible dimension (N×1). Note that we addeda nonnegativity condition in the optimization problem. This is done sothat the vector r have a spectral factor, i.e.,${{{r(0)} + {\sum\limits_{n = 1}^{N_{tap} - 1}{2{r(n)}{\cos \left( {2\pi \quad f\quad n\quad T_{2}} \right)}}}} \geq 0},$

for all frequencies. However, due to discretization this is notguaranteed.

We solve the LP to obtain the optimal auto-correlation vector r. Oncesuch a vector is obtained, the filter coefficients are then computedusing spectral factorization. For problems with small Nap_(tap),spectral factors can be obtained via root-finding methods. For example,see X. Chen and T. Parks, “Design of optimal minimum phase FIR filtersby direct factorization,” Signal Processing, 10:369-383, 1986.

FIGS. 6-12 illustrate one embodiment of the system of the presentinvention.

The transmit path for the HDSL2 system with all the relevant componentsis shown in FIG. 2. The output of the precoder (not shown) x₁(nT₁) isinput to the up-sampler 22. The sample rate of the input signal R=517 ⅓ksamples/sec (T₁=1/R sec). Let${{X_{1}(f)}\overset{\Delta}{=}{\sum\limits_{n = {- \infty}}^{\infty}{{x_{1}\left( {n\quad T_{1}} \right)}{\exp \left( {{- {j2}}\quad \pi \quad f\quad n\quad T_{1}} \right)}}}},$

be the frequency response of the input signal x₁(nT₁). This input signalif then up-sampled by a factor of 2 using zero-filling. Thus, the outputof the up-sampler is given by${x_{2}\left( {n\quad T_{2}} \right)}\overset{\Delta}{=}\left\{ \begin{matrix}{x_{1}\left( {m\quad T_{1}} \right)} & {{{{if}\quad n} = {2m}},} \\0 & {{otherwise},}\end{matrix} \right.$

where T₂=T₁/2. In the frequency domain we have $\begin{matrix}{{{X_{2}(f)}\quad \overset{\Delta}{=}{\sum\limits_{n = {- \infty}}^{\infty}{{x_{2}\left( {n\quad T_{2}} \right)}{\exp \left( {{- {j2}}\quad \pi \quad f\quad n\quad T_{2}} \right)}}}},} \\{\quad {{= {\sum\limits_{n = {- \infty}}^{\infty}{{x_{2}\left( {n\quad {T_{1}/2}} \right)}{\exp \left( {{- {j2}}\quad \pi \quad f\quad {T_{1}/2}} \right)}}}},}} \\{\quad {= {{X_{1}(f)}.}}}\end{matrix}$

The output of the up-sampler is then filtered by the transmit filterTxfil. Let the frequency response of the transmit filter be${{H(f)}\overset{\Delta}{=}{\sum\limits_{n = 0}^{N_{tap} - 1}{{h\left( {n\quad T_{2}} \right)}{\exp \left( {{- {j2}}\quad \pi \quad f\quad n\quad T_{2}} \right)}}}},$

where h(nT₂),n=0,1, . . . ,N_(tap)−1, are the filter coefficients andN_(tap) is the maximum allowable number of taps. We need to design thesecoefficients such that the PSD of the output x₇(t) satisfies the powerrequirements.

Using the filter response H(f), we write the output of the transmitfilter in the frequency domain as $\begin{matrix}{{{X_{3}(f)}\quad \overset{\Delta}{=}{\sum\limits_{n = {- \infty}}^{\infty}{{x_{3}\left( {n\quad T_{2}} \right)}{\exp \left( {{- {j2}}\quad \pi \quad f\quad n\quad T_{2}} \right)}}}},} \\{\quad {{= {{H(f)}{X_{2}(f)}}},}} \\{\quad {{= {{H(f)}{X_{1}(f)}}},}}\end{matrix}$

The output is then up-sampled again by a factor of 2 using zero-fillingand the output of the second up-sampler is given by${x_{4}\left( {n\quad T_{3}} \right)}\overset{\Delta}{=}\left\{ \begin{matrix}{x_{2}\left( {m\quad T_{2}} \right)} & {{{{if}\quad n} = {2m}},} \\0 & {{o\quad t\quad h\quad e\quad r\quad w\quad i\quad s\quad e},}\end{matrix} \right.$

where T₃=T₂/2=T,/4, and in the frequency domain we have $\begin{matrix}{{{X_{4}(f)}\quad \overset{\Delta}{=}{\sum\limits_{n = {- \infty}}^{\infty}{{x_{4}\left( {n\quad T_{3}} \right)}{\exp \left( {{- {j2}}\quad \pi \quad f\quad n\quad T_{3}} \right)}}}},} \\{\quad {{= {\sum\limits_{n = {- \infty}}^{\infty}{{x_{3}\left( {n\quad {T_{2}/2}} \right)}{\exp \left( {{- {j2}}\quad \pi \quad f\quad n\quad {T_{2}/2}} \right)}}}},}} \\{\quad {{= {{H(f)}{X_{1}(f)}}},}} \\{\quad {= {X_{3}(f)}}}\end{matrix}$

The output of the second up-sampler x₄(nT₃) is filtered through aninterpolating (low pass) filter H₁(f) (which sits inside the FPGA). Forthe up-stream we have chosen an 8^(th) order interpolating filter withthe following coefficients

h ₁(nT ₃)=[0.052277 0.26772 0.6602 1 1 0.6602 0.26772 0.052277],

and for the down-stream we have chosen a 5^(th) order interpolatingfilter with the following coefficients

h ₁(nT ₃)=[0.19811 0.6949 1 0.6949 0.19811],

The frequency response of these two filters are shown in FIG. 6. Theoutput of the interpolating filter in the frequency domain is given by$\begin{matrix}{{{X_{5}(f)}\quad \overset{\Delta}{=}{\sum\limits_{n = {- \infty}}^{\infty}{{x_{5}\left( {n\quad T_{3}} \right)}{\exp \left( {{- {j2}}\quad \pi \quad f\quad n\quad T_{3}} \right)}}}},} \\{\quad {{= {{H_{1}(f)}{X_{4}(f)}}},}} \\{\quad {{= {{H_{1}(f)}{H(f)}{X_{1}(f)}}},}}\end{matrix}$

The output of the interpolating filter is then passed through a sampleand hold (S/H) circuit followed by an analog filter H_(a)(f). The outputof the S/H is given by${{x_{6}(t)}\overset{\Delta}{=}{x_{5}\left( {nT}_{3} \right)}},\quad {{nT}_{3} \leq t < {\left( {n + 1} \right){T_{3}.}}}$

Thus, $\begin{matrix}{{{X_{6}(f)}\overset{\Delta}{=}\quad {\int\limits_{- \infty}^{\infty}{{x_{6}(t)}{\exp \left( {{- {j2}}\quad \pi \quad f\quad t} \right)}{t}}}},} \\{{= \quad {\sum\limits_{n = {- \infty}}^{\infty}{\int\limits_{n\quad T_{3}}^{{({n + 1})}T_{3}}{{x_{6}(t)}{\exp \left( {{- {j2}}\quad \pi \quad f\quad t} \right)}{t}}}}},} \\{{= \quad {\sum\limits_{n = {- \infty}}^{\infty}{\int\limits_{n\quad T_{3}}^{{({n + 1})}T_{3}}{{x_{5}\left( {n\quad T_{3}} \right)}{\exp \left( {{- {j2}}\quad \pi \quad f\quad t} \right)}{t}}}}},}\end{matrix}$

Where we have defined${H_{s\quad h}\overset{\Delta}{=}{{T_{3}{\exp \left( {{- j}\quad \pi \quad f\quad T_{3}} \right)}\frac{\sin \left( {\pi \quad f\quad T_{3}} \right)}{\pi \quad f\quad T_{3}}} = {T_{3}{\exp \left( {{- j}\quad \pi \quad f\quad T_{3}} \right)}\sin \quad {c\left( {f\quad T_{3}} \right)}}}},$

The magnitude plot the function H_(sh) is shown in FIG. 7. Now, we write

X ₆(f)=H _(sh) H ₁(f)H(f)X ₁(f).${= {\sum\limits_{n = {- \infty}}^{\infty}{{x_{5}\left( {n\quad T_{3}} \right)}{\int\limits_{n\quad T_{3}}^{{({n + 1})}T_{3}}{{\exp \left( {{- {j2}}\quad \pi \quad f\quad t} \right)}{t}}}}}},{= {{\left\lbrack {\sum\limits_{N = {- \infty}}^{\infty}{{x_{5}\left( {n\quad T_{3}} \right)}{\exp \left( {{- {j2}}\quad \pi \quad f\quad n\quad T_{3}} \right)}}} \right\rbrack \frac{1 - {\exp \left( {{- {j2}}\quad \pi \quad f\quad T_{3}} \right)}}{{j2}\quad \pi \quad f}} = {{X_{5}(f)}T_{3}{\exp \left( {{- j}\quad \pi \quad f\quad n\quad T_{3}} \right)}\frac{\sin \left( {\pi \quad f\quad T_{3}} \right)}{\pi \quad f\quad T_{3}}}}},{= {{X_{5}(f)}{H_{s\quad h}(f)}}},$

For the analog filter H_(a)(f). We have chosen a 4^(th) orderButterworth filter with cut-off frequency f_(c)=300 kHz for theup-stream and an 8^(th) order Butterworth filter with cut-off frequencyf_(c)=440 kHz kHz for the down-stream. The squared magnitude response ofan Nth order Butterworth filter with cut-off frequency f_(c) is given by$\left| {H_{a}(f)} \right|^{2} = {\frac{1}{1 + \left( {f/f_{c}} \right)^{2N}}.}$

The magnitude response of the up-stream and down-stream analog filtersare shown in FIG. 8.

Finally, the frequency response of x₇(t), the output of the analogfilter H_(a)(f) is given by $\begin{matrix}{{{X_{7}(f)}\overset{\Delta}{=}\quad {\int\limits_{- \infty}^{\infty}{{x_{7}(t)}{\exp \left( {{- {j2}}\quad \pi \quad f\quad t} \right)}\quad {t}}}},} \\{{= \quad {{H_{a}(f)}{X_{6}(f)}}},} \\{{= \quad {{H_{a}(f)}{H_{s\quad h}(f)}{H_{1}(f)}{H(f)}{X_{1}(f)}}},}\end{matrix}$

The HDSL2 standard specifies the PSD of x₇(t) in terms of dBm/Hz and thepower is measured with a load impedance of R_(L)=135Ω. Now the one-sidedPSD S_(x) ₇ (f) and the magnitude response X₇(f) are related as follows${{S_{x_{7}}(f)} = {\frac{2}{R_{L}f_{s}}{{X_{7}(f)}}^{2}}},$

where f_(s) is the sampling frequency in Hz. Thus,${S_{x_{7}}(f)} = {\frac{2}{R_{L}f_{s}}{{{H_{a}(f)}{H_{sh}(f)}{H_{1}(f)}{H(f)}}}^{2}{{{X_{1}(f)}}^{2}.}}$

Using the techniques described above, a thirty-two coefficient digitaltransmit filter for the upstream and downstream directions are shown inthe Table of FIG. 10.

FIG. 11 illustrates the overall downstream design PSD and specificationPSD, assuming |X(f)²=1.

FIG. 12 illustrates both the designed and specified overall upstreampower spectral densities.

Note that the transmit filter coefficients depend upon the otherelements in the transmit path since the other elements affect the W(f)frequency domain weighting function. Thus, for example, if a differentinterpolating filter were used, the coefficients of the transmit filterwould have to be modified.

It will be appreciated by those of ordinary skill in the art that theinvention can be implemented in other specific forms without departingfrom the spirit or character thereof. The presently disclosedembodiments are therefore considered in all respects to be illustrativeand not restrictive. The scope of the invention is illustrated by theappended claims rather than the foregoing description, and all changesthat come within the meaning and range of equivalents thereof areintended to be embraced herein.

What is claimed is:
 1. A digital transmit filter for a communicationsystem having a maximum prescribed output power spectral density, thedigital transmit filter having filter coefficients selected such thatthe output power spectral density of the system is below the maximumoutput power spectral density of the communication system, the filtercoefficients being selected from optimized autocorrelation coefficientsfor the filter in a convex optimization procedure.
 2. The digitaltransmit filter of claim 1 wherein spectral factorization is used todetermine the filter coefficients from the autocorrelation coefficients.3. The digital transmit filter of claim 2 wherein spectral factors ofthe spectral factorization are found using a root finding method.
 4. Thedigital transmit filter of claim 1 wherein the system includes aninterpolating filter that affects affects the total output powerspectral density of the system.
 5. The digital transmit filter of claim1 wherein a discrete approximization of an upper bound constraint isdone in the convex optimization procedure.
 6. The digital transmitfilter of claim 1 wherein the digital transmit filter is implementedwith different coefficients for upstream and downstream directions. 7.The digital transmit filter of claim 1 wherein the digital transmitfilter has fewer than one hundred coefficients.
 8. The digital transmitfilter of claim 1 wherein the digital transmit filter has thirty-two orfewer coefficients.
 9. A method of determining filter coefficients for adigital transmit filter of a communication system having a maximumprescribed output power spectral density, comprising optimizingautocorrelation coefficients for the digital transmit filter in a convexoptimization procedure; and determining the filter coefficients from theautocorrelation coefficients.
 10. The method of claim 9 wherein thefilter coefficients are determined from the autocorrelation coefficientsusing spectral factorization.
 11. The method of claim 10 wherein thespectral factorization is done using a root finding method.
 12. Themethod of claim 9 wherein a discrete approximization of an upper boundconstraint of the autocorrelation coefficients is done in the convexoptimization procedure.
 13. The method of claim 9 wherein the digitaltransmit filter is for a DSL system.
 14. The method of claim 13 whereinthe digital transmit filter is for a high-bit DSL system.
 15. The methodof claim 9 wherein the digital transmit filter has fewer than onehundred coefficients.
 16. The method of claim 9 wherein the digitaltransmit filter has thirty-two or fewer coefficients.